Grigory Garkusha (Swansea): Framed correspondences, part I

Framed correspondences were invented and studied by Voevodsky in
the early 2000-s, aiming at the construction of a new model for motivic
stable homotopy theory. Joint with Ivan Panin we introduce and study framed
motives of algebraic varieties basing on Voevodsky's framed correspondences.
Framed motives allow to construct an explicit model for the suspension
P1-spectrum of an algebraic variety. Framed correspondences
also give a kind of motivic infinite loop space machine. They also lead to
several important explicit computations such as rational motivic homotopy
theory or recovering the celebrated Morel theorem that computes certain
motivic homotopy groups of the motivic sphere spectrum in terms of
Milnor-Witt K-theory. In these lectures we shall discuss basic facts on
framed correspondences and related constructions.